The Annals of Statistics

Regularization and the small-ball method I: Sparse recovery

Guillaume Lecué and Shahar Mendelson

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Abstract

We obtain bounds on estimation error rates for regularization procedures of the form \begin{equation*}\hat{f}\in\mathop{\operatorname{argmin}}_{f\in F}(\frac{1}{N}\sum_{i=1}^{N}(Y_{i}-f(X_{i}))^{2}+\lambda \Psi(f))\end{equation*} when $\Psi$ is a norm and $F$ is convex.

Our approach gives a common framework that may be used in the analysis of learning problems and regularization problems alike. In particular, it sheds some light on the role various notions of sparsity have in regularization and on their connection with the size of subdifferentials of $\Psi$ in a neighborhood of the true minimizer.

As “proof of concept” we extend the known estimates for the LASSO, SLOPE and trace norm regularization.

Article information

Source
Ann. Statist., Volume 46, Number 2 (2018), 611-641.

Dates
Received: February 2016
Revised: January 2017
First available in Project Euclid: 3 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1522742431

Digital Object Identifier
doi:10.1214/17-AOS1562

Mathematical Reviews number (MathSciNet)
MR3782379

Zentralblatt MATH identifier
06870274

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 62G08: Nonparametric regression
Secondary: 62C20: Minimax procedures 62G05: Estimation 62G20: Asymptotic properties

Keywords
Empirical processes high-dimensional statistics

Citation

Lecué, Guillaume; Mendelson, Shahar. Regularization and the small-ball method I: Sparse recovery. Ann. Statist. 46 (2018), no. 2, 611--641. doi:10.1214/17-AOS1562. https://projecteuclid.org/euclid.aos/1522742431


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Supplemental materials

  • Supplementary material to “Regularization and the small-ball method I: sparse recovery”. In the supplementary material we study a general $X$ without assuming it is isotropic.